Projectile Motion
Projectile motion is the two-dimensional motion of an object launched into the air that moves under the influence of gravity alone, following a curved (parabolic) trajectory. The horizontal and vertical components of motion are independent: the horizontal component is uniform (constant velocity), while the vertical component is uniformly accelerated by gravity (g ≈ 9.8 m/s²). Galileo first described this decomposition in the early 17th century.
Key Formula
R = (v₀² × sin(2θ)) / g
LaTeX: R = \frac{v_0^2 \sin 2\theta}{g}
| Symbol | Meaning | Unit |
|---|---|---|
| R | Horizontal range | m |
| v₀ | Initial speed of launch | m/s |
| θ | Angle of launch above horizontal | degrees or radians |
| g | Acceleration due to gravity (9.8 m/s²) | m/s² |
Worked Example
Problem
A cricket ball is hit at 20 m/s at an angle of 30° above the horizontal. Calculate the range and maximum height reached. (g = 9.8 m/s², ignore air resistance.)
Solution
Step 1: Resolve initial velocity: v₀ₓ = 20 cos30° = 20 × 0.866 = 17.32 m/s; v₀ᵧ = 20 sin30° = 20 × 0.5 = 10 m/s. Step 2: Time of flight: T = 2v₀ᵧ/g = 2×10/9.8 ≈ 2.04 s. Step 3: Range: R = v₀ₓ × T = 17.32 × 2.04 ≈ 35.3 m. Step 4: Maximum height: H = v₀ᵧ² / (2g) = 100 / (2×9.8) ≈ 5.1 m.
Answer
Range ≈ 35.3 m; Maximum height ≈ 5.1 m.
Horizontal and vertical components during projectile motion
| Component | Initial value | Acceleration | Equation | Notes |
|---|---|---|---|---|
| Horizontal (x) | v₀ cosθ | 0 | x = v₀ cosθ × t | Constant velocity |
| Vertical (y) | v₀ sinθ | −g (downward) | y = v₀ sinθ × t − ½gt² | Uniformly decelerated then accelerated |
| Speed at peak | v₀ cosθ | g acts only vertically | vₓ = v₀ cosθ | Minimum speed of trajectory |
| Vertical velocity at peak | 0 | — | vᵧ = 0 | Defines maximum height |
Interactive Tools
PhET Projectile Motion Simulation
Launch projectiles at varying angles and speeds and observe trajectories
Open ToolDesmos Graphing Calculator
Plot parabolic trajectories by graphing x(t) and y(t) parametrically
Open ToolKhan Academy — Projectile Motion
Video series covering horizontal launch and angled projectile problems
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Free Fall
Free fall is the motion of an object under the influence of gravity alone, with no other forces such as air resistance acting on it. Near Earth's surface, all freely falling objects experience the same constant downward acceleration g ≈ 9.8 m/s², regardless of their mass — a result first demonstrated experimentally by Galileo Galilei. Free fall is a special case of uniformly accelerated motion.
Velocity
Velocity is the rate of change of displacement with respect to time, making it a vector quantity with both magnitude (speed) and direction. Average velocity equals total displacement divided by total time, while instantaneous velocity is the derivative of position with respect to time. Velocity is central to Newton's laws and is measured in metres per second (m/s).
Acceleration
Acceleration is the rate of change of velocity with respect to time, and is a vector quantity. An object accelerates whenever its speed changes, its direction changes, or both simultaneously. Acceleration is caused by a net force (Newton's second law) and is measured in metres per second squared (m/s²).
From Latin "projectilis" (thrown forward), from "proicere" — "pro-" (forward) + "iacere" (to throw). Galileo's "Two New Sciences" (1638) first provided the mathematical description of parabolic projectile trajectories.